Spin excitations in magnetic structures of different dimensions Wulf Wulfhekel Physikalisches Institut, Universität Karlsruhe (TH) Wolfgang Gaede Str. 1, D-76131 Karlsruhe
0. Overview Chapters of spin excitation 1. Why are excitations of any importance? 2. Excitations of ferromagnets in the Heisenberg model 3. Excitations of antiferromagnets in the Heisenberg model 4. Spin waves in bulk, thin films and stripes 5. Itinerant magnetism 6. Experimental techniques to study excitations questions
1. Why are magnetic excitations of any importance? Magnetic data storage
1. Why are magnetic excitations of any importance? Write poles and GMR sensors
1. Why are magnetic excitations of any importance? Reading (and writing) data from a disk Typical data speed: 120MB/sec = 1GHz
1. Why are magnetic excitations of any importance? Superparamagnetism A single domain particle with e.g. uniaxial magnetic anisotropy due to magnetocrystalline anisotropy or a elongated shape (shape anisotropy) has two states with minimal energy. In case the energy barrier given by the anisotropy cannot be overcome thermally within a certain time, the magnetic moment is stable. In case the barrier can be overcome, the magnetic moment flips randomly between the states and the particle becomes superparamagnetic.
1. Why are magnetic excitations of any importance? The magnetic moment of a bound electron Magnetic moment of ring current (orbital moment) 2 =− e 2 =− e 2m ℏ l =− B l = I A =− e r 2m m r l B = e ℏ − 24 J / T Bohr magneton 2m = 9.27 × 10 Magnetic moment of spin (spin moment) S =− B g s Landé factor of the electron g = 2.0023 ≈ 2 s ≫ l In bulk spin moment usually dominates Attention: The magnetic moment behaves like an angular moment (precession).
1. Why are magnetic excitations of any importance? Dynamic of magnetization reversal Ground state of magnetic particle is single domain. µMAG standard problem #4 NIST, Maryland (VA) USA, M. Donahue et al . http://www.ctcms.nist.gov/~rdm/mumag.org.html
1. Why are magnetic excitations of any importance? Magnetization dynamics +1 m y 0 -1 µ 0 H= 25mT � =170 ° <m x > φ <m y > <m z > During switcheing the partcile is not Single domain anymore. The magnetistatic energy is converted to spin waves.
0. Overview Chapters of spin excitation 1. Why are excitations of any importance? 2. Excitations of ferromagnets in the Heisenberg model 3. Excitations of antiferromagnets in the Heisenberg model 4. Spin waves in bulk, thin films and stripes 5. Itinerant magnetism 6. Experimental techniques to study excitations questions
2. Excitations in ferromagnets in the Heisenberg model Direct exchange interaction between two electrons Quantum mechanical system with two electrons : total wave function must be antisymmetric under exchange of the two electrons, as electrons are fermions. 1,2 =− 2,1 Wave function of electron is a product of spatial and spin part: 1 = r 1 × 1 1,2 = 1 For antiparallel spins (singlet): (↑↓ - ↓↑) antisymmetric 2 1 1 , 2 For parallel spins (triplet) : = ↑↑, (↑↓ + ↓↑), ↓↓ symmetric 2 → Spatial part of wave function has opposite symmetry to spin part
2. Excitations in ferromagnets in the Heisenberg model Direct exchange interaction between two electrons r 1, r 2 = 1 2 a r 1 b r 2 a r 2 b r 1 symmetric for singlet r 1, r 2 = 1 antisymmetric for triplet 2 a r 1 b r 2 − a r 2 b r 1 For the antisymmetric wave function : r 1, r 2 =− r 2, r 1 r ,r = 0 In case r 1 =r 2 follows : → Coulomb repulsion is lower for antisymmetric spatial wave function and thus its energy Is lower than that of the symmetrical spatial wave function Exchange interaction between two spins: difference of the coulomb energy due to symmetry 2 e * r 1 b * r 2 E S − E T = 2 ∫ a 4 0 ∣ r 1 − r 2 ∣ a r 2 b r 1 dr 1 dr 2
2. Excitations in ferromagnets in the Heisenberg model Direct exchange between localized electrons J = E S − E T , E ex =− 2J S 1 S 2 2 J>0 : parallel spins are favoured (ferromagnetic coupling) J<0: antiparallel spins are favoured (antiferromagnetic coupling) N J ij S i E =− ∑ Heisenberg model for N spins: S j i,j=1 As electrons are assumed as localized, wave functions decay quickly and mainly nearest neighbors contribute to exchange. E =− ∑ J S i S j Nearest neighbor Heisenberg model: i,j NN
2. Excitations in ferromagnets in the Heisenberg model Ferromagnetism J>0 Spins align in parallel at T=0 Elements : Fe, Co, Ni, Gd … Oxides : Fe 2 O 3 , CrO … Semiconductors : GaMnAs, EuS ... Above Curie temperature Tc, they become paramagnetic. Fe 1043K EuS 16.5K T C ≈ 2 z J ex J J 1 Co 1394K GaMnAs ca. 180K 3k B Ni 631K Gd 289K z nearest neighbors
2. Excitations in ferromagnets in the Heisenberg model Solving the excitation spectrum Quantum mechanically exact solution is extremely hard if not impossible. N coupled atoms of spin S have a (2S+1) N dimensional Hilbert space. Example: A 3x3x3 Fe cluster (S=2) has 5 27 = 745.058.059.623.827.125 states. Let us try in a 1D chain of atoms (S=1/2) with only nearest neighbor interactions: 1 + H =− 2 J ∑ z S i 1 + S i 1 - S i 1 S j =− 2J ∑ S i + = S x iS y z - S S i 2 S i S i j=i+1 i - = S x − iS y S + | 1 + | − 1 2 > = | 1 S 2 > = 0 S 2 > - | 1 2 > = | − 1 - | − 1 2 > = 0 S 2 > S
2. Excitations in ferromagnets in the Heisenberg model Solving the excitation spectrum 2 | > H | > =− 2NJS For the ground state (say S i =+S) : Naïve try for an excited state: .... .... | j > = j-th spin is flipped j 2 2JS 2 | j > − SJ | j 1 > − SJ | j − 1> H | j > = 2 − NJS The single flipped spin is no eigenstate of the Hamiltonian!
2. Excitations in ferromagnets in the Heisenberg model Solving the excitation spectrum for a 1D ferromagnetic chain Solution: excited states are described by the ground state plus small excitations named magnons (quasiparticles) that do not interact. Shortcomings: in reality, magnons do interact. H =− 2 J ∑ S i S j j=i+1 z ≈ S S j Semiclassical ansatz: See blackboard! x = A e i qja − t S j 8JS y = B e i qja − t S j ℏ = 4JS 1 − cos qa Solution: a
2. Excitations in ferromagnets in the Heisenberg model Magnons or spin waves q = 2 Excitation of spin 1
2. Excitations the ferromagnets in the Heisenberg model Magnons Lindis Pass, NZ
2. Excitations in ferromagnets in the Heisenberg model Magnons in ferromagnets fcc Co 2 q 2 = D q 2 Dispersion : ℏ = 4JS 1 − cos qa ≈ 2JS a D is called spin wave stiffness and behaves like an inverse mass Fe Co Ni D [meVÅ 2 ] 281 500 364 from Blundell
0. Overview Chapters of spin excitation 1. Why are excitations of any importance? 2. Excitations of ferromagnets in the Heisenberg model 3. Excitations of antiferromagnets in the Heisenberg model 4. Spin waves in bulk, thin films and stripes 5. Itinerant magnetism 6. Experimental techniques to study excitations questions
3. Excitations in antiferromagnets in the Heisenberg model Antiferromagnetism J<0 Spins align antiparallel at T=0 Elements : Mn, Cr … Oxides : FeO, NiO … Semiconductors : URu 2 Si 2 … Salts : MnF 2 ... Above Néel temperature T N , they become paramagnetic. Cr 297K FeO 198K NiO 525K
3. Excitations in antiferromagnets in the Heisenberg model Antiferromagnetic configurations Depending on the crystal structure, many different antiferromagnetic configurations may exist.
3. Excitations in antiferromagnets in the Heisenberg model Magnon dispersion of antiferromagnets Solution: two ferromagnetic sublattices that couple antiferromagnetically. H =− 2 J ∑ S i S j , J 0 j=i+1 z = S , S 2p 1 z Ansatz: S 2p =− S − 4JS See blackboard! ℏ =− 4 JS ∣ sin qa ∣ Solution: a
3. Excitations in antiferromagnets in the Heisenberg model Magnones in antiferromagnets RbMnF 3 ℏ =− 4JSsin qa ≈− 4 JSa q = v q Dispersion : v is called spin wave velocity; magnons behave like massless objects from Kittel
3. Excitations in antiferromagnets in the Heisenberg model Spinons Ground state Excited state Excited state .... .... .... .... .... .... ΔS=1 ΔS=1/2 Magnon Spinon KCuF 3 Magnon= 2 * Spinon From Helmholtz Center Berlin
0. Overview Chapters of spin excitation 1. Why are excitations of any importance? 2. Excitations of ferromagnets in the Heisenberg model 3. Excitations of antiferromagnets in the Heisenberg model 4. Spin waves in bulk, thin films and stripes 5. Itinerant magnetism 6. Experimental techniques to study excitations questions
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